Normal approximation for total magnetization in subcritical long-range Ising models
Abstract
We establish Berry--Esseen bounds and 1-Wassershtein bounds for the total spin of strictly subcritical zero-field Ising models with summable interactions. Let J:ℤd→[0,∞) be symmetric, satisfy J(0)=0, and |J|1:=∑x∈ℤdJ(x)<∞. For a nonempty finite set Λ⊂ℤd, consider zero-field Gibbs measure with free boundary conditions, in which a spin configuration σ∈{−1,+1}Λ has probability proportional to exp { β∑x,y∈Λ J(x−y)σxσy }. Let SΛ:=∑x∈Λσx be the total spin and βc denote the critical inverse temperature. For every fixed 0≤β<βc, we prove that dK ( SΛ/√(Var(SΛ)),G ) ≤ Cβ,J|Λ|−1/2, where dK is the Kolmogorov distance, G∼ 𝒩(0,1), and Cβ,J depends only on β and J. For the fully periodized model on (ℤ/Lℤ)d, the corresponding rate is L−d/2, uniformly in L. For the classical nearest-neighbor model, the Kolmogorov bound also holds for −βc<β<βc, covering both ferromagnetic and antiferromagnetic interactions. We also obtain $1$-Wasserstein bounds with the same rates in the respective subcritical regimes. Both the Kolmogorov and $1$-Wasserstein bounds hold with the same rate under the corresponding infinite-volume free-boundary Gibbs measures.
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BibTeX
@misc{liu2026normal,
title = {{Normal approximation for total magnetization in subcritical long-range Ising models}},
author = {Song-Hao Liu and Qi-Man Shao and Yang Xie and Jing-Yu Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22835},
eprint = {2610.22835},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22835},
}
RIS
TY - UNPB
TI - Normal approximation for total magnetization in subcritical long-range Ising models
AU - Song-Hao Liu
AU - Qi-Man Shao
AU - Yang Xie
AU - Jing-Yu Xu
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.22835
UR - https://arxiv.org/abs/2610.22835
AB - We establish Berry--Esseen bounds and 1-Wassershtein bounds for the total spin of strictly subcritical zero-field Ising models with summable interactions. Let J:ℤ^d→[0,∞) be symmetric, satisfy J(0)=0, and |J|_1:=∑_x∈ℤ^dJ(x)<∞. For a nonempty finite set Λ⊂ℤ^d, consider zero-field Gibbs measure with free boundary conditions, in which a spin configuration σ∈{−1,+1}^Λ has probability proportional to exp { β∑_x,y∈Λ J(x−y)σ_xσ_y }. Let S_Λ:=∑_x∈Λσ_x be the total spin and β_c denote the critical inverse temperature. For every fixed 0≤β<β_c, we prove that d_K ( S_Λ/√(Var(S_Λ)),G ) ≤ C_β,J|Λ|^−1/2, where d_K is the Kolmogorov distance, G∼ 𝒩(0,1), and C_β,J depends only on β and J. For the fully periodized model on (ℤ/Lℤ)^d, the corresponding rate is L^−d/2, uniformly in L. For the classical nearest-neighbor model, the Kolmogorov bound also holds for −β_c<β<β_c, covering both ferromagnetic and antiferromagnetic interactions. We also obtain $1$-Wasserstein bounds with the same rates in the respective subcritical regimes. Both the Kolmogorov and $1$-Wasserstein bounds hold with the same rate under the corresponding infinite-volume free-boundary Gibbs measures.
ER -